The singular Riemann–Roch theorem and Hilbert–Kunz functions

Abstract

In the paper, via the singular Riemann–Roch theorem, it is proved that the class of the eth Frobenius power A e can be described using the class of the canonical module ω A for a normal local ring A of positive characteristic. As a corollary, we prove that the coefficient β ( I , M ) of the second term of the Hilbert–Kunz function ℓ A ( M / I [ p e ] M ) of e vanishes if A is a Q -Gorenstein ring and M is a finitely generated A-module of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the eth Frobenius power F ∗ e O X can be described using the canonical divisor K X .